Abstract interpretation of domain-specific embedded languages
A domain-specific embedded language (DSEL) is a domain-specific programming language with no concrete syntax of its own. Defined as a set of combinators encapsulated in a module, it borrows the syntax and tools (such as type-checkers and compilers) of its host language; hence it is economical to design, introduce, and maintain. Unfortunately, this economy is counterbalanced by a lack of room for growth. DSELs cannot match sophisticated domain-specific languages that offer tools for domainspecific error-checking and optimisation. These tools are usually based on syntactic analyses, so they do not work on DSELs. Abstract interpretation is a technique ideally suited to the analysis of DSELs, due to its semantic, rather than syntactic, approach. It is based upon the observation that analysing a program is equivalent to evaluating it over an abstract semantic domain. The mathematical properties of the abstract domain are such that evaluation reduces to solving a mutually recursive set of equations. This dissertation shows how abstract interpretation can be applied to a DSEL by replacing it with an abstract implementation of the same interface; evaluating a program with the abstract implementation yields an analysis result, rather than an executable. The abstract interpretation of DSELs provides a foundation upon which to build sophisticated error-checking and optimisation tools. This is illustrated with three examples: an alphabet analyser for CSP, an ambiguity test for parser combinators, and a definedness test for attribute grammars. Of these, the ambiguity test for parser combinators is probably the most important example, due to the prominence of parser combinators and their rather conspicuous lack of support for the well-known LL(k) test. In this dissertation, DSELs and their signatures are encoded using the polymorphic lambda calculus. This allows the correctness of the abstract interpretation of DSELs to be proved using the parametricity theorem: safety is derived for free from the polymorphic type of a program. Crucially, parametricity also solves a problem commonly encountered by other analysis methods: it ensures the correctness of the approach in the presence of higher-order functions.